Very interesting. Tom is the Smith expert, and I also have never seen that damascus pattern. A Quality No. 0 or 1 of that era would be expected to have 'English Stub Twist' or possibly Two Iron Crolle that Hunter Arms called 'Good Damascus-Figured Pattern'

From a DE Remington


And very similar barrels from a Three Barrel Gun Co.


As Dave said, Pieper P from an EE Remington


Very likely those are special order Belgian damascus, and I have no way to confirm as yet, but they may be "Leclerc", which was a pattern like "Bernard" and came in several grades. Rene Leclerc was a barrel maker in Paris at the same time as Leopold Bernard.

https://sites.google.com/a/damascusknowledge.com/www/home